A guaranteed deterministic approach to superhedging: most unfavorable scenarios of market behaviour and moment problem

Abstract

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure the coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game-theoretical interpretation implies that the corresponding Bellman-Isaac equations hold, both for pure and mixed strategies. In the present paper, we propose a two-step method of solving the Bellman equation arising in the case of (game) equilibrium. In particular, the most unfavorable strategies of the `market can be found in the class of the distributions concentrated at most in n+1 point, where n is the number of risky assets.